User alexander alldridge - MathOverflow

Answer by Alexander Alldridge for References for Harish-Chandra pairs and modules, category "O"?

2012-03-23T08:28:31Z

This answer may be a little late, considering the date it was asked, but anyway...

Concerning question 4), a "standard" reference is perhaps the two volume treatise by Wallach on "Reductive Groups". Some aspects of Harish-Chandra modules (with a view towards globalisation questions) are also developed in a recent preprint by Bernstein and Kroetz (see B. Kroetz's web page). Jantzen's Habilitationsschrift has already been mentioned. If you can read German, this also an excellent source.

Over the last few years, Penkov, Serganova and Zuckermann have been developing the theory of (g,k)-modules in the general setting where g is some Lie algebra and k is some subalgebra. The modules one considers are g-modules which are semi-simple over k (plus possibly additional conditions). Check the list of recent preprints and publications of these authors.

Answer by Alexander Alldridge for Morphisms between supermanifolds R^{0|1}→R^{0|1}

2011-08-18T13:38:23Z

@Ma: As an answer to your following question:

Could you give a clue how to calculate the $map$, for example, $map(R^{0∣d},M)$ for any supermanifold $M$?

Take a look at arXiv:math/0307303, where this question is discussed.

For $d=1$, it is well-known (and due to Kontsevich, I think), that $map(R^{0|1},M)$ is the total space of the odd tangent bundle $\Pi TM$ of $M$.

If $U$ is a superdomain of dimension $p|q$, then $map(R^{0|d},U)=U\times R^{pr+qs|ps+qr}$ where $(r+1)|s=2^{d-1}|2^{d-1}$ is the graded dimension of $\bigwedge R^d$. This you can check using the definition of $map$ and the characterisation of morphisms of supermanifolds as given in Leites.

Another good source on this subject (for $d=1$), is the paper "Differential forms and 0-dimensional supersymmetric field theories" by Hohnhold, Kreck, Stolz and Teichner.

Answer by Alexander Alldridge for Gluing of manifolds and the Hausdorff condition.

2011-08-18T13:23:11Z

I am leaving this as an answer since I am new here and therefore can't comment on answers. It's not an answer, since I am not saying anything about the nerve of the covering. Apologies to the regulars here!

Here's what I wanted to say: David Carchedi's answer to the question is rather nice. However, it can be stated in much simpler terms, as follows.

Notice that the condition on the map $\coprod U_{\alpha\beta}\to\coprod U_\alpha\times\coprod U_\beta$ to be proper is simply that it be closed, since it is injective and its domain is Hausdorff. Moreover, it's open, so it's a homeomorphism onto its image, so it is closed if and only if its range is closed. The range of the map is just the graph of the equivalence relation $R$ on $X':=\coprod U_\alpha$ such that $X'/R$ is the glued space $X$.

Thus the statement is that $X$ is Hausdorff if and only $R$ has a closed graph in $X'\times X'$. This is true for any equivalence relation $R$ such that the canonical projection $X'\to X'/R=X$ is open. You may find this in Chapter 1, § 8.3 of Bourbaki, General Topology, vol. 1. The type of equivalence relation $R$ coming from such gluings as considered always has this property. This is also in Bourbaki, somewhere in § 4. Anyway, these things are not difficult to check by hand.